Injectivity of NURBS curves. (English) Zbl 1334.65047
Summary: The injectivity of NURBS curve implies the curve has no self-intersection. In this paper, we propose a geometric condition on the control polygon which guarantees the NURBS curve to be injective for all possible choices of positive weights. The proof is based on the degree elevation algorithm and toric degeneration theory of NURBS curve.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 68U07 | Computer science aspects of computer-aided design |
References:
| [1] | Piegl, L.; Tiller, W., The NURBS Book (1997), Springer: Springer New York · Zbl 0868.68106 |
| [2] | Piegl, L.; Tiller, W., Curve and surface constructions using rational B-spline, Comput. Aided Des., 19, 9, 485-498 (1987) · Zbl 0655.65012 |
| [3] | Piegl, L., On NURBS: A Survey, IEEE Comput. Graph., 1, 55-71 (1991) |
| [4] | Farin, G., Algorithms for rational Bézier curves, Comput. Aided Des., 15, 2, 73-77 (1983) |
| [5] | Hoffmann, C. M., Geometric and Solid Modeling (1989), Morgan Kaufmann: Morgan Kaufmann San Francisco |
| [6] | Müller, S.; Feliu, E.; Regensburger, G.; Conradi, C.; Shiu, A.; Dicikenstein, A., Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry, Found. Comput. Math., 1-29 (2013) |
| [7] | Lasser, D., Calculating the self-intersections of Bézier curves, Comput. Ind., 12, 3, 259-268 (1989) |
| [8] | Tiller, W.; Hanson, E., Offsets of two-dimensional profiles, IEEE Comput. Graph., 4, 9, 36-46 (1984) |
| [10] | Ravi Kumar, G.; Shastry, K.; Prakash, B., Computing non-self-intersection offsets of NURBS surfaces, Comput. Aided Des., 34, 209-228 (2002) |
| [11] | Xu, Z.; Ye, X.; Chen, Z.; Zhang, Y.; Zhang, S., Trimming self-intersections in swept volume solid modeling, J. Zhejiang Univ.-Sci. A, 9, 470-480 (2008) · Zbl 1140.74409 |
| [12] | Goodman, T.; Unsworth, K., Injective bivariate maps, Ann. Numer. Math., 3, 91-104 (1994) · Zbl 0873.65005 |
| [13] | Choi, Y.; Lee, S., Injectivity conditions of 2D and 3D uniform cubic B-spline functions, Graph. Models, 62, 6, 411-427 (2000) · Zbl 1011.68574 |
| [14] | Craciun, G.; Feinberg, M., Multiple equilibria in complex chemical reaction networks: I. The injectivity property, SIAM J. Appl. Math., 65, 5, 1526-1546 (2005) · Zbl 1094.80005 |
| [15] | Craciun, G.; García-Puente, L.; Sottile, F., Some geometrical aspects of control points for toric patches, (Dählen, M.; Floater, M. S.; Lyche, T.; Merrien, J. L.; Mørken, K.; Schumaker, L. L., Mathematical Methods for Curves and Surfaces (2010), Springer: Springer Berlin, Heidelberg), 111-135 · Zbl 1274.65033 |
| [17] | Zhu, C. G.; Zhao, X. Y., Self-intersections of rational Bézier curves, Graph Models, 76, 5, 312-320 (2014) |
| [18] | Zhao, X. Y.; Zhu, C. G., Injectivity conditions of rational Bézier surfaces, Comput. Graph., 51, 17-25 (2015) |
| [20] | Krishnamurthy, A.; Khardekar, R.; McMains, S.; Haller, K.; Elber, G., Performing efficient NURBS modeling operations on the GPU, IEEE Trans. Vis. Comput. Graphics, 15, 4, 530-543 (2009) |
| [21] | Krasauskas, R., Toric surface patches, Adv. Comput. Math., 17, 1-2, 89-113 (2002) · Zbl 0997.65027 |
| [22] | Farin, G., Curves and Surfaces for CAGD: A Practical Guide (2002), Morgan Kaufmann: Morgan Kaufmann San Francisco |
| [23] | Farin, G.; Hoschek, J.; Kim, M., Handbook of Computer Aided Geometric Design (2002), Elsevier: Elsevier Amsterdam · Zbl 1003.68179 |
| [24] | Davis, P., Interpolation and Approximation (1975), Dover Publications: Dover Publications New York · Zbl 0329.41010 |
| [25] | García-Puente, L.; Sottile, F.; Zhu, C. G., Toric degenerations of Bézier patches, ACM Trans. Graph., 30, 5, 110 (2011) |
| [26] | Piegl, L.; Tiller, W., Software-engineering approach to degree elevation of B-spline curves, Comput. Aided Des., 26, 17-28 (1994) · Zbl 0816.65002 |
| [27] | Prautzsch, H., Degree elevation of B-spline curves, Comput. Aided Geom. Design, 1, 193-198 (1984) · Zbl 0552.65011 |
| [28] | Cohen, E.; Lyche, T.; Schumaker, L., Algorithms for degree-raising of splines, ACM Trans. Graph., 4, 3, 171-181 (1985) · Zbl 0591.65012 |
| [29] | Prautzsch, H.; Piper, B., A fast algorithm to raise the degree of B-spline curves, Comput. Aided Geom. Design, 8, 253-266 (1991) · Zbl 0753.65008 |
| [30] | Wang, G.; Deng, C., On the degree elevation of B-spline curves and corner cutting, Comput. Aided Geom. Design, 24, 2, 90-98 (2007) · Zbl 1171.65321 |
| [31] | Nairn, D.; Peters, J.; Lutterkort, D., Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon, Comput. Aided Geom. Design, 16, 7, 613-631 (1999) · Zbl 0997.65016 |
| [32] | Karavelas, M.; Kaklis, P.; Kostas, K., Bounding the distance between 2D parametric Bézier curves and their control polygon, Computing, 72, 1-2, 117-128 (2004) · Zbl 1062.65017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
